Question

Use a determinant to determine whether the points are collinear. $$ (-1,-7),(0,-3),(1,2) $$

Short answer

The points (-1,-7), (0,-3), (1,2) are not collinear.

Step-by-step solution

Write the Coordinates in a 3x3 Matrix

Write the coordinates in a 3x3 matrix with last column being column of ones. The matrix will be \[ \begin{bmatrix} -1 & -7 & 1 \ 0 & -3 & 1 \ 1 & 2 & 1 \end{bmatrix} \]

Calculate the Determinant

Calculate the determinant of the matrix. The process involves diagonal multiplication and subtraction. Applying these rules, the determinant is calculated as follows: \[ = (-1)*(-3)*1 + (-7)*1*1 + 1*0*1 - 1*(-3)*1 - (-7)*1*1 - (-1)*0*1\ = 3 - 7 - 0 + 3 - (-7) -0\ = 6+7\ = 13 \]

Check the Determinant

Check if the determinant is equal to 0. This will confirm if the points are collinear or not. As the determinant \(13 \neq 0\), points are not collinear.

Key concepts

Determinant Calculation

Understanding the determinant of a matrix is crucial for various applications in linear algebra, including the assessment of collinearity among points.

Consider the determinant as a scalar attribute of a square matrix. It can be interpreted as a measure of the volume scaling factor that the matrix represents when it is used to transform space. Determinants are calculated from the elements of the matrix via a specific arithmetic procedure. For a 3x3 matrix, as in our case, the determinant can be found by a method known as the rule of Sarrus, or more generally by cofactor expansion.

In the given exercise, the matrix was constructed by placing the x and y coordinates of each point as the first and second columns, respectively, and filling the third column with ones. This allows the determinant to reflect the geometric properties of the points in question. If the calculation of the determinant yields zero, it means that the transformation associated with the matrix doesn't alter the volume, which in the context of points on a plane translates to them being on the same line.

Collinearity Test

The concept of collinearity in geometry refers to the alignment of points along a single straight line. This characteristic is quite important as it helps in understanding the geometrical relationships between points in a two-dimensional space.

To test for collinearity using algebra, one can employ a determinant-based method. When we arrange the coordinates of the points into a matrix and calculate the determinant, a result of zero indicates that the area of the triangle formed by the points is zero. Hence, no actual triangle exists and the points must be collinear. In contrast, a non-zero determinant signifies that the points form a triangle with a non-zero area, proving that they are not collinear.

Application in the Given Exercise

In the exercise provided, the determinant was found to be 13, which is clearly not zero. This non-zero value indicates that the triangle formed by connecting the three points has an area, implying that the points are not collinear. This method is a powerful tool in analyzing geometrical arrangements of points and can be efficiently applied to any number of dimensions, provided the determinant can be computed.

Matrix Algebra

Matrix algebra is a field within mathematics that deals with the study and manipulation of matrices. Matrices are arrays of numbers arranged in rows and columns and are used to solve systems of linear equations, to represent linear transformations, and in areas like physics, computer science, and statistics.

Key operations in matrix algebra include addition, subtraction, multiplication, and finding the inverse of a matrix, if it exists. Each operation follows specific rules and has practical implications for solving problems. For instance, matrix multiplication is not commutative, meaning the order in which matrices are multiplied affects the result.

Significance in Determinant Calculation and Collinearity

As for the exercise in question, matrix algebra is central to understanding how the determinant calculation is derived from the matrix representing points. Understanding matrix operations also sheds light on interpreting the results from operations involving the determinant, such as the collinearity test among points. Matrices help transform abstract geometrical concepts into tangible calculations that can be easily evaluated for concrete answers.